Tenney Height

If p/q is a positive rational number reduced to its lowest terms, then the Benedetti height is the integer pq. Often it is more convenient instead to take the logarithm, usually base 2 (log2), of the Benedetti height, leading to Tenney height. In either form it is widely used as a measure of inharmonicity and/or complexity for intervals.

The Tenney height of a monzo is given by

|| |e2 e3 ... ep> || = |e2| + log2(3)|e3| + ... + log2(p)|ep| = log2(2^|e2| * 3^|e3| * ... * p^|ep|)

Examples

Interval name	Frequency ratio	monzo	log2(Benedetti height)
unison	1/1	|0>	0
octave	2/1	|1>	1
just perfect fifth	3/2	|-1 1>	log2(6) = 2.585
just major third	5/4	|-2 0 1>	log2(20) = 4.322
harmonic seventh	7/4	|-2 0 0 1>	log2(28) = 4.807